# Foundations of Universal Numbers A fundamental extension beyond traditional number systems, unifying the properties of real, complex, and p-adic numbers within a computable framework consistent with UOR axioms. Universal numbers represent a fundamental extension beyond traditional number systems, unifying the properties of real, complex, and p-adic numbers within a computable framework while maintaining complete consistency with the UOR prime-coordinate axioms. The Universal Number system is built upon the following axioms, which extend the Core UOR Axioms to numerical domains: Axiom 1: Universal Prime Coordinate Representation Every universal number η ∈ 𝕌 admits a unique representation as an infinite tuple of prime exponents: η = φ(η) = (e₁, e₂, e₃, ...) where eᵢ ∈ ℂ represents the "exponent" of the i-th prime in the universal factorization of η. Axiom 2: Coordinate Homomorphism Extension The prime-coordinate homomorphism extends naturally to universal numbers, preserving the fundamental relationship: φ(η₁ · η₂) = φ(η₁) + φ(η₂) where addition in the right-hand side occurs in the infinite-dimensional complex vector space indexed by primes. Axiom 3: Completeness Under Multiple Topologies The field 𝕌 is simultaneously complete under both the Archimedean (complex) metric and all non-Archimedean (p-adic) metrics, with a coherent transition mechanism between these topological structures. Axiom 4: Computational Realizability Every universal number admits an effective algorithm for approximation to arbitrary precision in both Archimedean and non-Archimedean contexts. Axiom 5: Canonical Embedding Principle For each traditional number system (ℕ, ℤ, ℚ, ℝ, ℂ, ℚₚ), there exists a canonical embedding into 𝕌 that preserves all relevant algebraic and topological structures. With these axioms established, we formally define the field of universal numbers: 𝕌 = {η | η admits a unique prime-coordinate representation φ(η) = (e₁, e₂, e₃, ...) that is computably approximable in all relevant metrics} The core mathematical operation underpinning universal numbers is the Universal Coordinate Transformation: Φ: 𝕌 → ℂ^∞ mapping each universal number to its infinite-dimensional complex coordinate vector. Universal numbers maintain specific relationships with conventional number systems. Natural numbers and integers embed naturally into 𝕌 via their prime factorizations. Rational numbers embed as quotients of integer representations, yielding coordinates that can include negative exponents. Real and complex numbers embed via convergent sequences of rational approximations. For each prime p, the p-adic numbers ℚₚ embed via specialized coordinate maps that preserve the p-adic valuation. Theorem 1 (Universal Representation): Every universal number η ∈ 𝕌 admits a canonical representation as a triple: η = (z, (aₚ)ₚ∈ℙ, φ(η)) where z ∈ ℂ is its complex value, aₚ ∈ ℤₚ is its p-adic expansion for each prime p, and φ(η) is its prime coordinate vector. Universal numbers exhibit crucial meta-theoretical properties: Category-Theoretic Universality, Model-Theoretic Completeness, Information-Theoretic Optimality, and Intrinsic Coordinate Invariance. ## References - [[uor-c-034|universal-number]] - [[uor-c-035|universal-prime-representation]] - [[uor-c-036|universal-coordinate-transformation]] - [[uor-c-037|universal-representation-theorem]] ## Metadata - **ID:** urn:uor:resource:universal-numbers-foundation - **Author:** UOR Framework - **Created:** 2025-04-22T00:00:00Z - **Modified:** 2025-04-22T00:00:00Z